Consider a mass$-$spring$-$damper system with input$-$output equation: $m{x}^{}(t)+b{x}^{}(t)+kx(t)=f(t).$ Assume a mass of $m=10kg$ and a damping of $b=100N\frac{s}{m}.$

a$.$ We want to find the spring constant $k$ that makes this system critically damped. Calculate it analytically, which is easy to do for this simple system.

b$.$ Now that you know the correct answer, let's practice using a root$-$locus diagram to calculate the same parameter. The root$-$locus method can be used with more complicated systems that are too difficult to tackle analytically. First, reformulate the characteristic equation in a form that will let you consider the characteristic equation's roots as $k$ is varied from zero to infinity. Then, use MATLAB's rlocus function to plot the root locus $($turn in your plot, and the m$-$file used to generate it$).$ Click on the location of the root$-$locus in the location that represents the critically damped response, and find the value of $k$ at that point. Turn in any additional calculations that were required. Remember, since this is a numerical method, you might not get the exact answer that you calculated in part $($a$),$ but it should be very close.